The Guardians Council had announced on 22 May that it had rejected the applications of more than 1,000 presidential hopefuls, narrowing the field to just six ahead of the 17 June first round.
Moin had called his exclusion from the race "unfair, unreasonable, and illegal."
Khamenei reportedly intervened after a request from parliamentary speaker Gholam-Ali Haddad-Adel.
The move is not unprecedented, and the supreme leader holds broad authority under the country's constitution, but the Guardians Council has the final say over who is allowed to run for office.
Khamenei asked the Guardians Council to reconsider the disqualification of nearly half of the 8,145 people seeking to compete for legislative seat ahead of the country's parliamentary voting in early 2004. On that occasion, the Guardians Council reinstated the candidacies of some 1,160 people.
Akbar Atri, a member of the Allameh faction of the Office for Strengthening Unity student organization, predicted in an interview with Radio Farda on 23 May that Moin would be reinstated pursuant to the Khamenei's letter.
Atri also expressed the hope that on election day Iranian women would express their displeasure over the council's rejection of all prospective female candidates for president. Nearly 90 women registered to be candidates and, according to Atri, the council spokesman said women do not have the power or intellectual strength to administer society.